Lec07 Process Optimisation

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Process Design & Simulation 2013

Lecture 7

Process Optimisation

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Identify and represent quantitatively trade-offs in processdesign

Develop understanding of role of optimisation in processdesign

Develop understanding of how type of optimisationproblem affects solution strategy

Define and identify a simple optimisation problem andsolve it

Intended Learning Outcomes

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Optimisation is intrinsic to design Design decisions should be based on good designs

o Experience and judgement are usefulo Rigorous optimisation is often needed

Operations including planning, scheduling, supply-chainmanagement and selection of operating conditions benefit significantly from optimisation

Optimisation in process design and

operation

Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9Lec 07 3


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Structural (topology)optimisation discrete optionso Technology selection, flowsheet

configuration, connections

Parameter optimisation

continuous variableso Design and operating variables

(conversion, purity, pressure,temperature...)

Performance evaluationo Which measure of performance (key

performance indicator, KPI;objective function)?

Process synthesis, design and optimisation

Projectdefinition

Synthesis

Simulation

Evaluation

Finalflowsheet

Parameteroptimisation

Structuraloptimisation

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The model a mathematical description of the key aspectsof the problem

The process performance must be evaluated a suitable measure of performance must be defined this is the objective function

We need to set some constraints the required outcomes provide problem specifications

feasibility (e.g. non-negative mole fractions) and external issues(e.g. related to safety) may be needed to arrive at meaningfulsolutions

The overall optimisation problem ...

select values for the degrees of freedom of the problem in order toachieve the best performance model, constraints and objective function may include both discrete

and continuous variables model, constraints and objective function may have linear or non-

linear formulation

Process optimisation: Key concepts

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1 Optimisation example: Heat exchanger design2 Problem formulation

3 Objective functions

4 Convexity

5 Solving single-variable optimisation problems

6 Multivariable optimisation7 Constrained optimisation

Outline

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Problem decomposition may make the problem simplero Need to ensure that optimum solution for one unit (sub-problem) is

not at the expense of another unit (sub-problem)o Often need to consider process system and interactions between

units Need good understanding of physics and chemistry

o Ensure that degrees of freedom and constraints are satisfactorilyaddressed

o Identify and include significant issues Need good understanding of problem context

o What performance indicator(s)?o What constraints? ... solution must be feasible and practical

Need to select mathematical formulationo linear, non-linear, mixed integer/continuous variables?o what optimisation algorithm?o what initial guess?

Setting up the problem

Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.4Lec 07 7


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Simulation is a prediction mathematical model relatesinputs, specifications, constraints, etc. to outputso To simulate, number of unknown variables must be equal to number

of independent equationso i.e. number of degrees of freedom is zero

In optimisation, there remain some degrees of freedomo no. of equations, equality constraints, etc. is fewer than number of

unknownso during optimisation, the values of the degrees of freedom (decision

variables or optimisation variables) are selectedo ... to maximise or minimise the objective functiono ... subject to constraintso typically, optimisation will require repeated simulation using different

values of the optimisation variables

Simulation vs. optimisation

(Degrees of freedom)

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PROCESS

Hot wastestream

PROCESS

Heatrecovery

How big should we make the heat exchanger ?

How much heat should be recovered ?

Design problem:

1 Optimisation example: Heat exchanger design1 Design problem

A hot stream is available for heat recovery in a process. A heat

exchanger design is needed.

Smith, 2005 , Chemical Process Design and Integration , Wiley, Ch. 3.1Lec 07 9


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Write equations (equality constraints) :

QREC = mH CP H (TH in - TH out ) -

QREC = m C CP C (TC out - TC in) -

Energy Cost = (Q H - QREC ) CE

-U TLMQRECHeat transfer area (A) =

Annualised capital cost = (a + bA c)AF - 5

- 4

1

3

2

Enthalpy change :

mH = mass flowrate of hot stream (kg.s 1)

mC = mass flowrate of cold stream (kg.s 1)

CP C = heat capacity of cold stream (kJ.kg 1 .K 1)

CP H = heat capacity of hot stream (kJ.kg 1 .K 1)

QH = hot utility demand without heat

recovery (kJ.s 1)CE = unit cost of energy ($.kJ 1 or $.kW 1 .h)

A = heat exchange area (m 2)

U =overall heat transfer coefficient (kJ.m 2 .K 1 s 1)

TLM

= logarithamic mean temperaturedifference (K)

a,b,c = cost coefficients

AF = annualisation factor

T = temperature (K)

Optimisation example:Heat exchanger design

2. Mathematical modelTH in TH out

TC in

TC out

Hot stream(being cooled)

Cold stream(being heated)

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In addition to above 13 known variables, there are 6 unknown variables

(QREC , TH out , TC out , Energy Cost, Annualised Capital Cost, A)

= 19 variables

Degrees of freedom= [No of variables] [No of equality constraints]= 19 18

There is one degree of freedom to be optimised

... Let us select heat recovery as the optimisation variable


3. Problem analysisFive Equations + Specifications for 13 variables

(mH, m

C,C

P H, C

P C, T

H in,T

C in, U, a, b, c, AF, Q

H, C

E)

= 18 equality constraints

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T

H Q REC

T C i n

T H out

T C out

Increase heatrecovery

T H in

Hot stream

Conditions in heat recovery exchanger

Optimisation example:

Heat exchanger design4. Graphical representation

TH in TH out

TC in

TC out

Hot stream(being cooled)

Cold stream(being heated)

Graphical representation can help with insights

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Trade-off existsbetween capital and

energy

Cost

EnergyCost

Total Cost

CapitalCost

Optimum Heat recovered


5. Application of model

Recovery of heat from a waste steam involves a trade-off betweenreduced energy cost and increased capital cost of heat exchanger.

Objective function : Total Annualised Cost (TAC) = Energy cost +Annualised Capital Cost

Smith, 2005, Chemical Process Design and Integration, Wiley, Ch. 3.1Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.3

Cost

EnergyCost

CapitalCost

Heat recovered

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Process optimisation problems typically involve discretedecisionso e.g. how many stages? which connection between equipment? how

many of a given type of unit?o these often translate into integer variables

2 Problem formulation

Integer vs. continuous variables

Hot stream 2 ?

Cold stream

Hot stream 3 ?

Cold stream

Hot stream 1

Cold stream

Example of discrete decision which heat recovery match? Other important process parameters form continuous variables

o e.g. what operating pressure? what volume of tank? what product purity? The overall problem formulation depends on that of the process

model, the objective function and the constraintsLec 07 14


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Most process optimisation problems are non-linearo i.e. model + objective function + constraintso e.g. physical properties models, reaction models, mass balance

using mole fractions, cost calculations...

Problem formulation Non-linear problems

Energy Cost = (Q H - QREC ) CE

U TLMQREC

Heat transfer area (A) =

Annualised capital cost = (a + bA c)AF

Cost

EnergyCost

Total Cost

CapitalCost


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It is often possible to formulatethe problem as a linear problemo model, constraints and objective

function must all be represented usinglinear (or piece-wise linear)expressions

o but... the accuracy of the modellingmay be compromised

The benefits of a linearformulation are that theoptimisation problems are mucheasier to solve

... and the optimum that is foundis guaranteed to be the globaloptimum

No. batches (product 2)

Constraint: Use of unit 2

Revenue dependent on no. of batches

Constraint: Use of unit 1

No. batches (product 1)

Smith, 2005, Chemical Process Design and Integration, Wiley, Ch. 3.5

Two products produced batch-wise intwo- step process

Value of products, time required in eachprocessing step known Constraint: Availability of equipment for

each processing step Objective: Maximise revenue from

production

Problem formulation Linear problems

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The objective function measures how good the design iso Is the design fit for purpose? How effectively does it satisfy customer needs?o Measure of performance is maximised or minimised

3 Objective function

Profit

Net present value Process yield Plant availability

Project expenditure

Cost of production Total annualised cost Waste production

Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.1

Process economics dominates decision makingo e.g. maximise profit Good understanding of the important issues can allow a simpler

objective to be seto e.g. maximise yield

Difficult to quantify some costs and benefitso e.g. health, environment, safety, societal impact

May need to consider uncertainties in prices, sales volumes, etc.Lec 07 17


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Process Design & Simulation 2013Not all objective functions are as straightforward

In the heat exchanger design example there was only one 'extremepoint (minimum) in the objective function i.e. it was unimodal .

Objective functionsExample: Unimodal objective functions

Cost

EnergyCost

Total Cost

CapitalCost


Smith, 2005 , Chemical Process Design and Integration ,Lec 07 18


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x

f(x)

Discontinuous function

DiscontinuityStationaryPoint

x

f(x)

Multimodal function

LocalOptimum

StationaryPoints

GlobalOptimum

LocalOptimum

Complex forms of objective functions

A 'false' optimum can be obtained, depending on where we start the searcho A local optimum is not the best solution

A zero gradient is a necessary but not sufficient condition for optimality



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If we draw a straight line between any two points on afunction...f(x)

x1 x2If the function is to beminimised and all valuesof the function lie belowthe straight line

convex function

x

f(x)

x1 x2x

If the function is to bemaximised and all valuesof the function lie abovethe straight line

concave function

4 Convexity and concavity



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Objective functions:Convex and concave functions

A convex or concave objective function provides a single optimumo If we find a minimum for a function that is to be minimised, and is known to

be convex , then we know it is the global optimumo

If we find a maximum for a function that is to be maximised, and is known tobe concave , then we know it is the global optimum Non-convex and non-concave functions can have local optima

Smith, 2005 , Chemical Process Design and Integration , Wiley, Ch. 3.1

f(x)

x1 x2x

convex function

f(x)

x1 x2x

neither concave nor convexconcave function

f(x)

x1 x2x

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x1

f(x)

x2 x

5 Solving single-variable optimisationproblems

1 Region elimination

If f(x1) > f(x2), we needsearch only for x > x 1

An example of a method for single variable search is regionelimination

The function is assumed to be unimodal

x1

f(x)

x2 x If f(x1) < f(x2), we need

search only for x < x 1

x1

f(x)

x2 x We keep narrowing the

search space until x 1 x2 i.e. x 1 and x 2 are within a

specified tolerance


f(x)

x

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f(x)

xx1

Evaluate derivative of objective functiongiven candidate solution, x 1

x2

Estimate new candidate solution, x 2, byassuming objective function is linear

x3

xOPTIterate until successive valuesof x are sufficiently close

Solving single-variable optimisation problems2 Newton's method

Method can be extremely efficient However, the solution procedure can be unstable if function is not

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Higher-dimensional optimisation problems

Single variable problem find highest point on line

Two-variable problem find mountain top

Two-variable problem find highest peak in

mountain range

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2001005020 Global

Optimum

Local Optimum

x2

x1

If the optimisation involves two variables, then we canrepresent it as a contour plot

Higher-dimensional optimisation problems

The concepts of convexity and concavity can be extended toproblems with more than one variable If a straight line between any two points on the surface always lies above (or

below) the surface, the function is concave (or convex, respectively)

Smith, 2005 , Chemical Process Design and Integration , Wiley, Ch. 3.1Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.7Lec 07 25


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6 Multivariable optimisation methods

Direct searcho do not require gradients

Indirect searcho use gradients to select search direction

Stochastic optimisationo use random choices to guide search

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e.g. univariate search (parametric search)

NOTE: Start at the wrong initialisation and we will find the

local optimum.

Multivariable optimisation methods1 Direct search methods

all variables except one arefixed remaining variable is

optimised

this variable is then fixedand another variable isoptimised, etc.


x2 2001005020

Startingpoint

x2

x1

GlobalOptimum

Local Optimum

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e.g. steepest descent (ascent)

Multivariable optimisation methods2 Indirect search methods

maximum rate of change of theobjective function gives searchdirection

problems caused by gradient

changing significantly duringsearch and choice of step size search can become extremely

slow as the optimum is

approached

Smith, 2005 , Chemical Process Design and Integration , Wiley, Ch. 3.3Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.7

NOTE : Start at the wrong initialisation and we will find thelocal optimum.

x2 2001005020

Startingpoint

x2

x1

GlobalOptimum

Local Optimum

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Multivariable optimisation methods3 Stochastic optimisation

All of the methods so far seek to improve the objective functionat each step.

Unfortunately, this can mean that the search is attracted towards a localoptimum.

Stochastic search methods generate a random path to thesolution based on probabilities .

Improvement in the objective function becomes the ultimategoal , rather than the immediate goal.

Some deterioration of the objective function is tolerated,

especially during the early stages of a search. Stochastic optimisation reduces the problem of becoming

trapped in a local optimum, and of requiring good initial values. ... usually at the expense of computation time

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Issues external to the design problem often constrain thesolutionso e.g. safety, environmental issues, materials, design codes and

standards, available resource, the marketo these may be represented mathematically as inequalities or equalities

A good understanding of the design problem and its contextis essential to formulate the problem constraintso ... otherwise the optimal solution may be far from practical or feasiblet

Sinnott and Towler, 2009, Chemical Engineering Des

7 Constraints in design

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Constrained optimisation

Most optimisation problems involve constraints General form of an optimisation problem involves three

basic elements:

1 An objective function to be optimised (e.g. minimisetotal cost, maximise economic potential, etc.).

2 Equality constraints, which are equations describing

the model of the process or equipment.3 Inequality constraints, expressing minimum or

maximum limits on various parameters.

Inequality constraints reduce the solution space to beexplored

In general, the existence of constraints complicates the

problem relative to the problem with no constraintsLec 07 31

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Constraints on solution space are imposed on objective function

Feasibleregion

x 2

x 1

Unconstrainedoptimum

Unconstrained optimum canbe reached (no constraintsactive at optimum)

x 2

x 1

Unconstrainedoptimum

Feasible region

Constrainedoptimum

Unconstrained maximumcannot be reached (constraintsactive at optimum)

x 2

x 1

UnconstrainedoptimumLocal

optimum

A non-convex region mightprevent the global optimumfrom being reached.

To ensure we reach the global optimum we need aconvex function and a convex region

Constrained optimisationGraphical representation


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The search for global optimality

If problem is linear, global optimality can be guaranteed If problem is non-linear, global optimality cannot be guaranteed Most design problems are non-linear

Solutions

Objective

Function

In the region of the optimum there are usually several solutions

with very similar performance Don't concentrate on one solution with the absolute lowest value of

the objective functiono always uncertainties in the datao many other factors to consider in the final design

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Design optimisation industrial practice


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Time constraints vs. value added Economic uncertainties are large Capital cost estimates are highly

approximate

Safety, operability, reliability etc.not embedded in optimisation Typically several near-optimal

solutions exist

Design optimisation industrial practice

Understanding of physical

phenomena is important! Which costs dominate? What are constraints and

causes of step changes(discontinuities)?

What trade-offs need to beaccounted for?

How sensitive is performance(objective) to important designparameters?

... need to develop confidencethat design is close to optimalIn practice, rigorousoptimisation is rare

Sinnott and Towler, 2009, Chemical Engineering Design , Ch. 1.9.11Lec 07 34

Summary


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Models allow quantitative, mathematical representation ofprocess, design objectives and constraints Optimisation employs systematic techniques to find the best

process designs

Optimisation can address fixed flowsheets and flowsheetgenerationo discrete design decisions typically involve integer variables

Which optimisation strategy is appropriate depends on the natureof the mathematical formulations structural vs. parametric variables, number of degrees of freedom, convexity

of problem and objective function

Typically, process design problems are non-linear 'mixed integer'(both discrete and continuous variables), constrained, non-convexproblemso local optima, infeasibilities and discontinuities present significant challenges

for process optimisation

Summary


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